Synthetic quantum Hall bilayer (SQHB), realized by optically driven monolayer graphene in the quantum Hall regime, provides a flexible platform for engineering quantum Hall phases as discussed in Ghazaryan et al. [Phys. Rev. Lett. 119, 247403 (2017)]. The coherent driving which couples two Landau levels mimics an effective tunneling between synthetic layers. The tunneling strength, the effective Zeeman coupling, and two-body interaction matrix elements are tunable by varying the driving frequency and the driving strength. Using infinite density matrix renormalization group techniques combined with exact diagonalization, we show that the system exhibits a non-Abelian bilayer Fibonacci phase at filling fraction nu = 2/3. Moreover, at integer filling nu = 1, the SQHB exhibits quantum Hall ferromagnetism. Using Hartree-Fock theory and exact diagonalization, we show that excitations of the quantum Hall ferromagnet are topological textures known as skyrmions.